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Article 24

On a theorem of Lee and Chew

Anal. St. Univ. Ovidius Constanta 5 (1996), no. 1, 51-58.

Mathematical reviews subject classification: 26A39, 26A45, 26A46.



Riesz and Nagy (see Leçons d'analyse fonctionelle, 3nd. ed., Akad. Kiado, Budapest, 1955) developed the Lebesgue integral on an interval, using the mean convergence of a step function (see also Gh. Marinescu, Analiza matematica, vol. 1, Editura Academiei Bucuresti, 1983 in Romanian).

Motivated by this work, Lee and Chew (see P. Y. Lee and T. S. Chew, A Riesz-type definition of the Denjoy integral, Real Analysis Exchange 11 (1985/6), 221-227;  P.Y. Lee, Lanzhou lectures on Henstock integration, World Scientific, Singapore, 1989) showed that:

A function f:[a,b] → R that is Denjoy*-integrable may be defined as a controlled convergent sequence of step functions.

However, their proof isn't complete (see Remark 4). 

The purpose of this paper is to show that this result is indeed true, and to give a complete proof.